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Conformal Geometry and Dynamics

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Mating Kleinian groups isomorphic to $C_2\ast C_5$ with quadratic polynomials

Author: Marianne Freiberger
Journal: Conform. Geom. Dyn. 7 (2003), 11-33
MSC (2000): Primary 37F45, 37F30, 37F05; Secondary 37F10
Published electronically: May 27, 2003
MathSciNet review: 1992035
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Abstract: Given a quadratic polynomial $q:\hat {\mathbb {C}}\rightarrow \hat {\mathbb {C}}$ and a representation $G:\hat {\mathbb {C}} \rightarrow \hat {\mathbb {C}}$ of $C_2\ast C_5$ in $PSL(2,\mathbb {C})$ satisfying certain conditions, we will construct a $4:4$ holomorphic correspondence on the sphere (given by a polynomial relation $p(z,w)$) that mates the two actions: The sphere will be partitioned into two completely invariant sets $\Omega$ and $\Lambda$. The set $\Lambda$ consists of the disjoint union of two sets, $\Lambda _+$ and $\Lambda _-$, each of which is conformally homeomorphic to the filled Julia set of a degree 4 polynomial $P$. This filled Julia set contains infinitely many copies of the filled Julia set of $q$. Suitable restrictions of the correspondence are conformally conjugate to $P$ on each of $\Lambda _+$ and $\Lambda _-$. The set $\Lambda$ will not be connected, but it can be joined up using a family $\mathcal {C}$ of completely invariant curves. The action of the correspondence on the complement of $\Lambda \cup \mathcal {C}$ will then be conformally conjugate to the action of $G$ on a simply connected subset of its regular set.

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Additional Information

Marianne Freiberger
Affiliation: School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, UK

Keywords: Holomorphic correspondences, rational maps, Kleinian groups, quasi-conformal surgery
Received by editor(s): November 23, 2001
Received by editor(s) in revised form: March 13, 2003
Published electronically: May 27, 2003
Additional Notes: The author was supported in part by the European Commission and EPSRC. The author would like to thank Shaun Bullett for many enlightening conversations. Also, thanks to the referee for useful comments
Article copyright: © Copyright 2003 American Mathematical Society